All of the 5th grade teachers and students from Santa Rita went on a field trip to an art museum. Tickets were $$7.00$ each for teachers and $$4.50$ each for students, and the group paid $$48.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$14.00$ each for teachers and $$9.50$ each for students, and the group paid $$99.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+4.5y = 48}$ ${14x+9.5y = 99}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-14x-9y = -96}$ ${14x+9.5y = 99}$ Add the top and bottom equations together. $ 0.5y = 3 $ $ y = \dfrac{3}{0.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {7x+4.5y = 48}$ to find $x$ ${7x + 4.5}{(6)}{= 48}$ $7x+27 = 48$ $7x = 21$ $x = \dfrac{21}{7}$ ${x = 3}$ You can also plug ${y = 6}$ into $ {14x+9.5y = 99}$ and get the same answer for $x$ ${14x + 9.5}{(6)}{= 99}$ ${x = 3}$ There were $3$ teachers and $6$ students on the field trips.